Lemma 3.1 (Theorem 2.2 in [1]})  
For each \( R\in \mathbb {N}\)  and \( \mu \) -a.s. \( \xi \) ,
the probability measure \( \mu _{R, \xi }^{m}\circ (\mathfrak {l} _{R}^{m})^{-1}\)  has a density
\( \mathbf {m}_{R, \xi }^{m}\)  with respect to the Lebesgue measure such that,
for \( \mathbf {x}= (x^i)_{i=1}^{m} \in \mathbf {S}_{R}^{m}\) ,
\(&c_{\ref {;A1}}^{-1}\Delta ^m(\mathbf {x}) \le \mathbf {m}_{R, \xi }^{m}(\mathbf {x}) \le c_{\ref {;A1}}\Delta ^m(\mathbf {x}).\) 
